West Ham went into the match on the fringe of the top four but were poor all afternoon and James Tomkins' injury-time header came far too late to make any difference.
Manager Martinez said: "Before the game we knew it was a must-win game, and that's always difficult to cope with. It brings extra expectations and all of a sudden you start wanting to win and you don't really know how to win.
"I thought today was the complete opposite. I thought the performance was very mature, we kept the ball really well at times, we were a real threat and probably the most pleasing aspect was the manner we coped with their attacking threat.
"Until the last minute we showed real composure and I was extremely pleased with that. It was a very well deserved three points and now is probably the start of our season."